How do I change the name of the default action created when inserting keyframe?
keyframe_insert() creates animation_data and an action if necessary, and makes it the new action of the active object.

Thus, you can simply access your object's action and change its name:

bpy.data.objects["Rig"].animation_data.action.name = "foobar"


Note that you shouldn't do this before any call to keyframe_insert(), as it would error out if there's either no animation_data or no action.

Refer to this question to learn about other approaches for animating things:

Python performance with Blender operators